Decimals: Simplification of Expressions

In this page, handling of decimals, both positive and negative decimals, in numerical expressions is explained.

The precedence order PEMDAS / BODMAS is explained.

For operations of same precedence order, the sequence of operation "simplification from left to right" is explained.

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Let us quickly revise what is numerical expression, and precedence order, sequence in simplifying the numerical expressions. This was introduced in whole numbers, reviewed in integers, and reviewed again in fractions. It takes very little time to revise the same in the context of decimals.

What is `2+4+3`?

`9`

`9`

`243`

The answer is "`9`".

This is an example of a numerical expression.

Which of the following is a meaning for the word "expression"?

something done very fast

collection of numbers and arithmetic operations between them, which together represent a quantity

collection of numbers and arithmetic operations between them, which together represent a quantity

The answer is "collection of numbers and arithmetic operations between them, which together represent a quantity".

What is `2xx4xx3`?

`24`

`24`

`243`

The answer is "`24`".

This is an example of a numerical expression.

Is `a+3x` a numerical expression?

Yes. It has the number 3.

No. It is not entirely numbers and arithmetic operations

No. It is not entirely numbers and arithmetic operations

The answer is "No. It is not entirely numbers and arithmetic operations".

Is `3+4-2+1` a numerical expression?

No, it has both addition and subtraction

Yes, addition and subtraction can be part of a numerical expression

Yes, addition and subtraction can be part of a numerical expression

The answer is "Yes, addition and subtraction can be part of an expression".

Is `3+4xx2-6-:3` a numerical expression?

No, as it has the addition, subtraction, multiplication, division

Yes, all arithmetic operations can be part of a numerical expression

Yes, all arithmetic operations can be part of a numerical expression

The answer is "Yes, all arithmetic operations can be part of a numerical expression"

is `3` a numerical expression?

No. `3` is only a number. Not a numerical expression

Yes. technically a number can also be considered a numerical expression

Yes. technically a number can also be considered a numerical expression

The answer is "Yes. technically a number can also be considered a numerical expression".

Consider `1+2` and `3xx1`. Note that when evaluated, both result in identical numerical value `1+2=3` and `3xx1 = 3`.

Are these two different numerical expressions?

They are two different expressions, evaluating to equal values

They are two different expressions, evaluating to equal values

They are identical expressions as they evaluate to equal values

The answer is "They are two different expressions, evaluating to equal values".

Simplify `9-6-:3`

`7`

`7`

`1`

The answer is "`7`". Division has higher precedence over subtraction. So

`9-6-:3`

`=9-2`

`=7`

What is the rule of precedence in numerical expressions?

multiplication and division have higher precedence over addition and subtraction

parentheses or brackets have the highest precedence

both the above

both the above

The answer is "both the above".

Simplify `20-4-3`?

`13`

`13`

`19`

The answer is "`13`".

Two or more operations in the same precedence level are performed from left to right sequence.

`20-4-3`

`= 16-3` `=13`

What is the rule of sequence in numerical expressions?

there is no such thing as rule of sequence

when multiple operations of same precedence is to be simplified, the operations are performed from left to right sequence

when multiple operations of same precedence is to be simplified, the operations are performed from left to right sequence

The answer is "when multiple operation of same precedence is to be simplified, the operations are performed from left to right sequence".

In a numerical expression, the precedence order is: • division and multiplication in same level same level of precedence • addition and subtraction in same level of precedence.

When multiple operation of same precedence is to be simplified, the operations are performed from left to right sequence.

All these were studied as part of whole numbers, integers, and fractions. The same applies for decimals.

• Precedence order BODMAS / PEMDAS

• Left to Right sequence for same precedence

Let us see some more expressions with decimals.

Simplify `0.1 - 0.06 -:.3`

`0.3`

`-0.1`

`-0.1`

The answer is "`-0.1`". Division has higher precedence over subtraction. So

`.1-.06-:.3`

`=.1-0.2`

`=-0.1`

Simplify `6-:.3xx0.5`

`40`

`10`

`10`

The answer is "`10`".

The division and multiplication are of same precedence, so it is simplified from left to right.

`6-:.3xx.5`

`=20xx0.5`

`=10`

Simplify `6-:(0.3xx0.5)`

`40`

`40`

`10`

The answer is "`40`".

The bracket has higher precedence, and so the expression inside bracket is simplified first.

`6-:(.3xx.5)`

`=6-:(.15)`

`=40`

Solved Exercise Problem:

Simplify `4+(-0.5)-:(0.25)xx(-2)`

`8`

`8`

`5`

The answer is "`8`".

The division and multiplication are of higher precedence over addition. so `(-0.5)-:(0.25)xx(-2)` is to be simplified first.

In that, the division and multiplication are of same precedence, so it is simplified from left to right.

`4+(-.5)-:(.25)xx(-2)`

`=4+(-2)xx(-2)`

`=4+4`

`=8`.

Solved Exercise Problem:

Simplify `.001-(-.0001)-2xx(-.0015)`.

`-0.0019`

`0.0041`

`0.0041`

The answer is "`0.0041`". The multiplication is of higher precedence over subtraction and so `2xx(-.0015)` is simplified first. Then the two subtraction are in the same precedence level and so they are simplified in the left to right sequence.

`.001-(-.0001)-2xx(-.0015)`

`.001-(-.0001)-(-.003)`

`=0.0011-(-.003)`

`=0.0041`

Solved Exercise Problem:

Simplify `(-1+1.5-1)xx2+(-1.5)-0.5`

`-3`

`-3`

`-1`

The answer is "`-3`"

`(-1+1.5-1)xx2+(-1.5)-0.5`

`=(0.5-1)xx2+(-1.5)-0.5`

`=-0.5 xx 2 +(-1.5)-0.5`

`=-1 +(-1.5)-0.5`

`=(-2.5) -0.5`

`=-3`

Solved Exercise Problem:

What is the value of `-3-((-.1)-2xx(.01))`?

`-3.12`

`-2.88`

`-2.88`

The answer is "`-2.88`".

`-3-((-.1)-2xx(.01))`

brackets take precedence and within the brackets, the multiplication is higher precedence. `=-3-((-0.1)-0.02)`

brackets have higher precedence `=-3-(-0.12)`

simplifying `=-2.88`

Solved Exercise Problem:

What is the value of `-1.5-(-.25)-0.25`?

`1.5`

`-1.5`

`-1.5`

The answer is "`-1.5`". `-1.5-(-.25)-0.25`

left to right sequence `=-1.25-0.25`

`=-1.5`

Solved Exercise Problem:

Simplify `0.8+0.75-:(-1.5)`

`-0.3`

`0.3`

`0.3`

The answer is "`.3`".

`0.8+0.75-:(-1.5)`

The division is of higher precedence over addition. `=0.8+(-.5)`

`=0.3`

Solved Exercise Problem:

Simplify `(10-3-2)xx(-0.2)`.

`-1`

`-1`

`-15`

The answer is "`-1`".

`(10-3-2)xx(-0.2)`

The expression inside bracket is simplified first. The two subtraction are in the same precedence level and so they are simplified in the left to right sequence. `=(7-2)xx(-.2)` `=5xx(-.2)`

`=-1`

Simplification of Expressions : BODMAS

• B - Brackets

• O - Order (exponents, roots, logarithm)

• D - Division

• M - Multiplication

• A - Addition

• S - Subtraction

• And Left to Right sequence for multiple operations of same precedence.

PEMDAS

• P - Parentheses

• E - Exponents (roots and logarithm)

• M - Multiplication

• D - Division

• A - Addition

• S - Subtraction

• And Left to Right sequence for multiple operations of same precedence.